RESISTANCE
, or Resisting Force, in Physics, any power which acts in opposition to another, so as to destroy or diminish its effect. |
There are various kinds of Resistance, arising from the various natures and properties of the resisting bodies, and governed by various laws: as, the Resistance of solids, the Resistance of fluids, the Resistance of the air, &c. Of each of these in their order, as below.
Resistance of Solids, in Mechanics, is the force with which the quiescent parts of solid bodies oppose the motion of others contiguous to them.
Of these, there are two kinds. The first where the resisting and the resisted parts, i. e. the moving and quiescent bodies, are only contiguous, and do not cohere; constituting separate bodies or masses. This Resistance is what Leibnitz calls Resistance of the surface, but which is more properly called friction: for the laws of which, see the article Friction.
The second case of Resistance, is where the resisting and resisted parts are not only contiguous, but cohere, being parts of the same continued body or mass. This Resistance was first considered by Galileo, and may properly be called renitency.
As to what regards the Resistance of bodies when struck by others in motion, see Percussion, and Collision.
Theory of the Resistance of the Fibres of Solid Bodies. —To conceive an idea of this Resistance, or renitency of the parts, suppose a cylindrical body suspended vertically by one end. Here all its parts, being heavy, tend downwards, and endeavour to separate the two contiguous planes or surfaces where the body is the weakest; but all the parts of them resist this separation by the force with which they cohere, or are bound together. Here then are two opposite powers; viz, the weight of the cylinder, which tends to break it; and the force of cohesion of the parts, which resists the fracture.
If now the base of the cylinder be increased, without increasing its length; it is evident that both the Resistance and the weight will be increased in the same ratio as the base; and hence it appears that all cylinders of the same matter and length, whatever their bases be, have an equal Resistance, when vertically suspended.
But if the length of the cylinder be increased, without increasing its base, its weight is increased, while the Resistance or strength continues unaltered; consequently the lengthening has the effect of weakening it, or increases its tendency to break.
Hence to find the greatest length a cylinder of any matter may have, when it just breaks with the addition of another given weight, we need only take any cylinder of the same matter, and fasten to it the least weight that is just sufficient to break it; and then consider how much it must be lengthened, so that the weight of the part added, together with the given weight, may be just equal to that weight, and the thing is done. Thus, let l denote the first length of the cylinder, c its weight, g the given weight the lengthened cylinder is to bear, and w the least weight that breaks the cylinder l, also x the length sought; then as l : x : : c : (cx)/l = the weight of the longest cylinder sought; and this, together with the given weight g, must be equal to c together with the weight w; hence then the whole length of the cylinder sought. If the cylinder must just break with its own weight, then is g = 0, and in that case is the whole length that just breaks by its own weight. By this means Galileo found that a copper wire, and of consequence any other cylinder of copper, might be extended to 4801 braccios or fathoms of 6 feet each.
If the cylinder be fixed by one end into a wall, with the axis horizontally; the force to break it, and its Resistance to fracture, will here be both different; as both the weight to cause the fracture, and the Resistance of the fibres to oppose it, are combined with the effects of the lever; for the weight to cause the fracture, whether of the weight of the beam alone, or combined with an additional weight hung to it, is to be supposed collected into the centre of gravity, where it is considered as acting by a lever equal to the distance of that centre beyond the face of the wall where the cylinder or other prism is fixed; and then the product of the said whole weight and distance, will be he momentum or force to break the prism. Again, the Resistance of the fibres may be supposed collected into the centre of the transverse section, and all acting there at the end of a lever equal to the vertical semidiameter of the section, the lowest point of that diameter being immoveable, and about which the whole diameter turns when the prism breaks; and hence the product of the adhesive force of the fibres multiplied by the said semidiameter, will be the momentum of Resistance, and must be equal to the former momentum when the prism just breaks.
Hence, to find the length a prism will bear, fixed so horizontally, before it breaks, either by its own weight, or by the addition of any adventitious weight; take any length of such a prism, and load it with weights till it just break. Then, put l = the length of this prism, c = its weight, w = the weight that breaks it, a = distance of weight w, g = any given weight to be borne, d = its distance, x = the length required to break.
Then l : x : : c :(cx)/l the weight of the prism x, and its momentum; also dg = the momentum of the weight g; therefore (cx2)/(2l) + dg is the momentum of the prism x and its added weight. In like manner 1/2cl + aw is that of the former or short prism and the weight that brake it; consequently is the length sought, that just | breaks with the weight g at the distance d. If this weight g be nothing, then is the length of the prism that just breaks with its own weight.
If two prisins of the same matter, having their bases and lengths in the same proportion, be suspended horizontally; it is evident that the greater has more weight than the lesser, both on account of its length, and of its base; but it has less Resistance on account of its length, considered as a longer arm of a lever, and has only more Resistance on account of its base; therefore it exceeds the lesser in its momentum more than it does in its Resistance, and consequently it must break more easily.
Hence appears the reason why, in making small machines and models, people are apt to be mistaken as to the Resistance and strength of certain horizontal pieces, when they come to execute their designs in large, by observing the same proportions as in the small.
When the prism, fixed vertically, is just about to break, there is an equilibrium between its positive and relative weight; and consequently those two opposite powers are to each other reciprocally as the arms of the lever to which they are applied, that is, as half the diameter to half the axis of the prism. On the other hand, the Resistance of a body is always equal to the greatest weight which it will just sustain in a vertical position, that is, to its absolute weight. Therefore, substituting the absolute weight for the Resistance, it appears, that the absolute weight of a body, suspended horizontally, is to its relative weight, as the distance of its centre of gravity from the fixed point or axis of motion, is to the distance of the centre of gravity of its base from the same.
The discovery of this important truth, at least of an equivalent to it, and to which this is reducible, we owe to Galileo. On this system of Resistance of that author, Mariotte made an ingenious remark, which gave birth to a new system. Galileo supposes that where the body breaks, all the sibres break at once; so that the body always resists with its whole absolute force, or the whole force that all its fibres have in the place where it breaks. But Mariotte, finding that all bodies, even glass itself, bend before they break, shews that fibres are to be considered as so many little bent springs, which never exert their whole force, till stretched to a certain point, and never break till entirely unbent. Hence those nearest the fulcrum of the lever, or lowest point of the fracture, are stretched less than those farther off, and consequently employ a less part of their force, and break later.
This consideration only takes place in the horizontal situation of the body: in the vertical, the fibres of the base all break at once; so that the absolute weight of the body must exceed the united Resistance of all its fibres; a greater weight is therefore required here than in the horizontal situation, that is, a greater weight is required to overcome their united Resistance, than to overcome their several Resistances one after another.
Varignon has improved on the system of Mariotte, and shewn that to Galileo's system, it adds the consideration of the centre of percussion. In each system, the section, where the body breaks, moves on the axis of equilibrium, or line at the lower extremity of the same section; but in the second, the fibres of this section are continually stretching more and more, and that in the same ratio, as they are situated farther and farther from the axis of equilibrium, and consequently are still exerting a greater and greater part of their whole force.
These unequal extensions, like all other forces, must have some common centre where they are united, making equal efforts on each side of it; and as they are precisely in the same proportion as the velocities which the several points of a rod moved circularly would have to one another, the centre of extension of the section where the body breaks, must be the same as its centre of percussion. Galileo's hypothesis, where fibres stretch equally, and break all at once, corresponds to the case of a rod moving parallel to itself, where the centre of extension or percussion does not appear, as being confounded with the centre of gravity.
Hence it follows, that the Resistance of bodies in Mariotte's system, is to that in Galileo's, as the distance of the centre of percussion, taken on the vertical diameter of the fracture, is to the whole of that diameter. Hence also, the Resistance being less than what Galileo imagined, the relative weight must also be less, and in the ratio just mentioned. So that, after conceiving the relative weight of a body, and its Resistance equal to its absolute weight, as two contrary powers applied to the two arms of a lever, in the hypothesis of Galileo, there needs nothing to change it into that of Mariotte, but to imagine that the Resistance, or the absolute weight, is become less, in the ratio above mentioned, every thing else remaining the same.
One of the most curious, and perhaps the most useful questions in this research, is to find what figure a body must have, that its Resistance may be equal or proportional in every part to the force tending to break it. Now to this end, it is necessary, some part of it being conceived as cut off by a plane parallel to the fracture, that the momentum of the part retrenched be to its Resistance, in the same ratio as the momentum of the whole is to its Resistance; these four powers acting by arms of levers peculiar to themselves, and are proportional in the whole, and in each part, of a solid of equal Resistance. From this proportion, Varignon easily deduces two solids, which shall resist equally in all their parts, or be no more liable to break in one part than in another: Galileo had found one before. That discovered by Varignon is in the form of a trumpet, and is to be fixed into a wall at its greater end; so that its magnitude or weight is always diminished in proportion as its length, or the arm of the lever by which its weight acts, is increased. It is remarkable that, howsoever different the two systems may be, the solids of equal Resistance are the same in both.
For the Resistance of a solid supported at each end, as of a beam between two walls, see Beam.
Resistance of Fluids, is the force with which | bodies, moving in fluid mediums, are impeded and retarded in their motion.
A body moving in a fluid is resisted from two causes. The first of these is the cohesion of the parts of the fluid. For a body, in its motion, separating the parts of a fluid, must overcome the force with which those parts cohere. The second is the inertia, or inactivity of matter, by which a certain force is required to move the particles from their places, in order to let the body pass.
The retardation from the first cause is always the same in the same space, whatever the velocity be, the body remaining the same; that is, the Resistance is as the space run through, in the same time: but the velocity is also in the same ratio of the space run over in the same time: and therefore the Resistance, from this cause, is as the velocity itself.
The Resistance from the second cause, when a body moves through the same fluid with different velocities, is as the square of the velocity. For, first the Resistance increases according to the number of particles or quantity of the fluid struck in the same time; which number must be as the space run through in that time, that is, as the velocity: but the Resistance also increases in proportion to the force with which the body strikes against every part; which force is also as the velocity of the body, so as to be double with a double velocity, and triple with a triple one, &c: therefore, on both these accounts, the Resistance is as the velocity multiplied by the velocity, or as the square of the velocity. Upon the whole therefore, on account of both causes, viz, the tenacity and inertia of the fluid, the body is resisted partly as the velocity and partly as the square of the velocity.
But when the same body moves through different fluids with the same velocity, the Resistance from the second cause follows the proportion of the matter to be removed in the same time, which is as the density of the fluid.
Hence therefore, if d denote the density of the fluid, v the velocity of the body, and a and b constant coefficients: then adv2 + bv will be proportional to the whole Resistance to the same body, moving with different velocities, in the same direction, through fluids of different densities, but of the same tenacity.
But, to take in the consideration of different tenacities of fluids; if t denote the tenacity, or the cohesion of the parts of the fluid, then adv2 + btv will be as the said whole Resistance.
Indeed the quantity of Resistance from the cohesion of the parts of fluids, except in glutinous ones, is very small in respect of the other Resistance; and it also increases in a much lower degree, being only as the velocity, while the other increases as the square of the velocity, and rather more. Hence then the term btv is very small in respect of the other term adv2; and consequently the Resistance is nearly as this latter term; or nearly as the square of the velocity. Which rule has been employed by most authors, and is very near the truth in slow motions; but in very rapid ones, it differs considerably from the truth, as we shall perceive below; not indeed from the omission of the small term ctv, due to the cohesion, but from the want of the full counter pressure on the hinder part of the body, a vacuum, either perfect or partial, being left behind the body in its motion; and also perhaps to some compression or accumulation of the fluid against the fore part of the body. Hence,
To conceive the Resistance of fluids to a body moving in them, we must distinguish between those fluids which, being greatly compressed by some incumbent weight, always close up the space behind the body in motion, without leaving any vacuity there; and those fluids which, not being much compressed, do not quickly fill up the space quitted by the body in motion, but leave a kind of vacuum behind it. These differences, in the resisting fluids, will occasion very remarkable varieties in the laws of their Resistance, and are absolutely necessary to be considered in the determination of the action of the air on shot and shells; for the air partakes of both these affections, according to the different velocities of the projected body.
In treating of these Resistances too, the fluids may be considered either as continued or discontinued, that is, having their particles contiguous or else as separated and unconnected; and also either as elastic or nonelastic. If a fluid were so constituted, that all the particles composing it were at some distance from each other, and having no action between them, then the Resistance of a body moving in it would be easily computed, from the quantity of motion communicated to those particles; for instance, if a cylinder moved in such a fluid in the direction of its axis, it would communicate to the particles it met with, a velocity equal to its own, and in its own direction, when neither the cylinder nor the parts of the fluid are elastic: whence, if the velocity and diameter of the cylinder be known, and also the density of the fluid, there would thence be determined the quantity of motion communicated to the fluid, which (as action and reaction are equal) is the same with the quantity lost by the cylinder, and consequently the Resistance would thus be ascertained.
In this kind of discontinued fluid, the particles being detached from each other, every one of them can pursue its own motion in any direction, at least for some time, independent of the neighbouring ones; so that, instead of a cylinder moving in the direction of its axis, if a body with a surface oblique to its direction be supposed to move in such a fluid, the motion which the parts of the fluid will hence acquire, will not be in the direction of the resisted body, but perpendicular to its oblique surface; whence the Resistance to such a body will not be estimated from the whole motion communicated to the particles of the fluid, but from that part of it only which is in the direction of the resisted body. In fluids then, where the parts are thus discontinued from each other, the different obliquities of that surface which goes foremost, will occasion considerable changes in the Resistance; although the transverse section of the solid should in all cases be the same: And Newton has particularly determined that, in a fluid thus constituted, the Resistance of a globe is but half the Resistance of a cylinder of the same diameter, moving, in the direction of its axis, with the same velocity.
But though the hypothesis of a fluid thus constituted | be of great use in explaining the nature of Resistances, yet we know of no such fluid existing in nature; all the fluids with which we are conversant being so formed, that their particles either lie contiguous to each other, or at least act on each other in the same manner as if they did: consequently, in these fluids, no one particle that is contiguous to the resisted body, can be moved, without moving at the same time a great number of others, some of which will be distant from it; and the motion thus communicated to a mass of the fluid, will not be in any one determined direction, but different in all the particles, according to the different positions in which they lie in contact with those from which they receive their impulse; whence, great numbers of the particles being diverted into oblique directions, the Resistance of the moving body, which will depend on the quantity of motion communicated to the fluid in its own direction, will be different in quantity from what it would be in the foregoing supposition, and its estimation becomes much more complicated and oper<*>se.
If the fluid be compressed by the incumbent weight of its upper parts (as all fluids are with us, except at their very surface), and if the velocity of the moving body be much less than that with which the parts of the fluid would rush into a void space, in consequence of their compression; it is evident, that in this case the space left by the moving body will be instantaneously filled up by the fluid; and the parts of the fluid against which the foremost part of the body presses in its motion, will, instead of being impelled forwards in the direction of the body, in some measure circulate towards the hinder part of the body, in order to restore the equilibrium, which the constant influx of the fluid behind the body would otherwise destroy; whence the progressive motion of the fluid, and consequently the Resistance of the body, which depends upon it, would in this instance be much less, than in the hypothesis where each particle is supposed to acquire, from the stroke of the resisting body, a velocity equal to that with which the body moved, and in the same direction. Newton has determined, that the Resistance of a cylinder, moving in the direction of its axis, in such a compressed fluid as we have here treated of, is but one-fourth part of the Resistance to the same cylinder, if it moved with the same velocity in a fluid constituted in the manner described in the first hypothesis, each fluid being supposed of the same density.
But again, it is not only in the quantity of their Resistance that these fluids differ, but also in the different manner in which they act upon solids of different forms moving in them. In the discontinued fluid, first described, the obliquity of the foremost surface of the moving body would diminish the Resistance; but the same thing does not hold true in compressed fluids, at least not in any considerable degree; for the chief Resistance in compressed fluids arises from the greater or less facility with which the fluid, impelled by the fore part of the body, can circulate towards its hinder part; and this being little, if at all, affected by the form of the moving body, whether it be cylindrical, conical, or spherical, it follows, that while the transverse section of the body is the same, and consequently the quan- tity of impelled fluid also, the change of figure in the body will scarcely affect the quantity of its Resistance.
And this case, viz, the Resistance of a compressed fluid to a solid, moving in it with a velocity much less than what the parts of the fluid would acquire from their compression, has been very fully considered by Newton, who has ascertained the quantity of such a Resistance, according to the different magnitudes of the moving body, and the density of the fluid. But he expressly informs us that the rules he has laid down, are not generally true, but only upon a supposition that the compression of the fluid be increased in the greater velocities of the moving body: however, some unskilful writers, who have followed him, overlooking this caution, have applied his determination to bodies moving with all sorts of velocities, without attending to the different compressions of the fluids they are resisted by; and by this means they have accounted the Resistance, for instance, of the air to musket and cannon shot, to be but about one-third part of what it is found to be by experience.
It is indeed evident that the resisting power of the medium must be increased, when the resisted body moves so fast that the fluid cannot instantaneously press in behind it, and fill the deserted space; for when this happens, the body will be deprived of the pressure of the fluid behind it; which in some measure balanced its Resistance, or at least the fore pressure, and must support on its fore part the whole weight of a column of the fluid, over and above the motion it gives to the parts of the same; and besides, the motion in the particles driven before the body, is less affected in this case by the compression of the fluid, and consequently they are less deflected from the direction in which they are impelled by the resisted surface; whence it happens that this species of Resistance approaches more and more to that described in the first hypothesis, where each particle of the fluid being unconnected with the neighbouring ones, pursued its own motion, in its own direction, without being interrupted or deflected by their contiguity; and therefore, as the Resistance of a discontinued fluid to a cylinder, moving in the direction of its axis, is 4 times greater than the Resistance of a fluid sufficiently compressed of the same density, it follows that the Resistance of a fluid, when a vacuity is left behind the moving body, may be near 4 times greater than that of the same fluid, when no such vacuity is formed; for when a void space is thus left, the Resistance approaches in its nature to that of a discontinued fluid.
This then may probably be the case in a cylinder moving in the same compressed fluid, according to the different degrees of its velocity; so that if it set out with a great velocity, and moves in the fluid till that velocity be much diminished, the resisting power of the medium may be near 4 times greater in the beginning of its motion than in the end.
In a globe, the difference will not be so great, because, on account of its oblique surface, its Resistance in a discontinued medium is but about twice as much as in one properly compressed; for its oblique surface diminishes its Resistance in one case, and not in the other: however, as the compression of the medium, | even when a vacuity is left behind the moving body, may yet confine the oblique motion of the parts of the fluid, which are driven before the body, and as in an elastic fluid, such as our air is, there will be some degree of condensation in those parts; it is highly probable that the Resistance of a globe, moving in a compressed fluid with a very great velocity, may greatly exceed the proportion of the Resistance to slow motions.
And as this increase of the resisting power of the medium will take place, when the velocity of the moving body is so great, that a perfect vacuum is left behind it, so some degree of augmentation will be sensible in velocities much short of this; for even when, by the compression of the fluid, the space left behind the body is instantaneously filled up; yet, if the velocity with which the parts of the fluid rush in behind, is not much greater than that with which the body moves, the same reasons that have been urged above, in the case of an absolute vacuity, will hold in a less degree in this instance; and therefore it is not to be supposed that, in the increased Resistance which has been hitherto treated of, it immediately vanishes when the compression of the fluid is just sufficient to prevent a vacuum behind the resisted body; but we must consider it as diminishing only according as the velocity, with which the parts of the fluid follow the body, exceeds that with which the body moves.
Hence then it may be concluded, that if a globe sets out in a resisting medium, with a velocity much exceeding that with which the particles of the medium would rush into a void space, in consequence of their compression, so that a vacuum is necessarily left behind the globe in its motion; the Resistance of this medium to the globe will be much greater, in proportion to its velocity, than what we are sure, from Sir I. Newton, would take place in a slower motion. We may farther conclude, that the resisting power of the medium will gradually diminish as the velocity of the globe decreases, till at last, when it moves with velocities which bear but a small proportion to that with which the particles of the medium follow it, the Resistance becomes the same with what is assigned by Newton in the case of a compressed fluid.
And from this determination may be seen, how false that position is, which asserts the Resistance of any medium to be always in the duplicate ratio of the velocity of the resisted body; for it plainly appears, by what has been said, that this can only be considered as nearly true in small variations of velocity, and can never be applied in comparing together the Resistances to all velocities whatever, without incurring the most enormous errors. See Robins's Gunnery, chap. 2 prop. 1, and my Select Exercises pa. 235 &c. See also the articles Resistance of the Air, Projectile, and Gunnery.
Resistance and retardation are used indifferently for each other, as being both in the same proportion, and the same Resistance always generating the same retardation. But with regard to different bodies, the same Resistance frequently generates different retardations; the Resistance being as the quantity of motion, and the retardation that of the celerity. For the difference and measure of the two, see Retardation.
The retardations from this Resistance may be com- pared together, by comparing the Resistance with the gravity or quantity of matter. It is demonstrated that the Resistance of a cylinder, which moves in the direction of its axis, is equal to the weight of a column of the fluid, whose base is equal to that of the cylinder, and its altitude equal to the height through which a body must fall in vacuo, by the force of gravity, to acquire the velocity of the moving body. So that, if a denote the area of the face or end of the cylinder, or other prism, v its velocity, and n the specific gravity of the fluid; then, the altitude due to the velocity v being (v2)/(4g), the whole Resistance, or motive force m, will be ; the quantity g being = 16 1/12 feet, or the space a body falls, in vacuo, in the first second of time. And the Resistance to a globe of the same diameter would be the half of this.—Let a ball, for instance, of 3 inches diameter, be moved in water with a celerity of 16 feet per second of time: now from experiments on pendulums, and on falling bodies, it has been found, that this is the celerity which a body acquires in falling from the height of 4 feet; therefore the weight of a cylinder of water of 3 inches diameter, and 4 feet high, that is a weight of about 12 lb 4 oz, is equal to the Resistance of the cylinder; and consequently the half of it, or 6 lb 2 oz is that of the ball. Or, the formula (anv2)/(4g) gives oz, or 12 lb 4 oz, for the Resistance of the cylinder, or 6 lb 2 oz for that of the ball, the same as before.
Let now the Resistance, so discovered, be divided by the weight of the body, and the quotient will shew the ratio of the retardation to the force of gravity. So if the said ball, of 3 inches diameter, be of cast iron, it will weigh nearly 61 ounces, or 3 4/5 lb; and the Resistance being 6 lb 2 oz, or 98 ounces; therefore, the Resistance being to the gravity as 98 to 61, the retardation, or retarding force, will be 98/61 or 1 3/5, the force of gravity being 1. Or thus; because a the area of a great circle of the ball, is = pd2, where d is the diameter, and p = .7854, therefore the Resistance to the ball is ; and because its solid content is , and its weight (2/3)Npd3, where N denotes its specific gravity; therefore, dividing the Resistance or motive force m by the weight w, gives the retardation, or retarding force, that of gravity being 1; which is therefore as the square of the velocity directly, and as the diameter inversely; and this is the reason why a large ball overcomes the Resistance better than a small one, of the same density. See my Select Exercises, pa. 225 &c.
Resistance of Fluid Mediums to the Motion of Falling Bodies.—A body freely descending in a fluid, is accelerated by the relative gravity of the body, (that is, the difference between its own absolute gravity and that of a like bulk of the fluid), which continually acts upon it, yet not equably, as in a vacuum: the Resistance of the fluid occasions a retardation, or diminution | of acceleration, which diminution increases with the velocity of the body. Hence it happens, that there is a certain velocity, which is the greatest that a body can acquire by falling; for if its velocity be such, that the Resistance arising from it becomes equal to the relative weight of the body, its motion can be no longer accelerated; for the motion here continually generated by the relative gravity, will be destroyed by the Resistance, or the force of Resistance is equal to the relative gravity, and the body forced to go on equably: for after the velocity is arrived at such a degree, that the resisting force is equal to the weight that urges it, it will increase no longer, and the globe must afterward continue to descend with that velocity uniformly. A body continually comes nearer and nearer to this greatest celerity, but can never attain accurately to it. Now, N and n being the specific gravities of the globe and fluid, N - n will be the relative gravity of the globe in the fluid, and therefore is the weight by which it is urged downward; also is the Resistance, as above; therefore these two must be equal when the velocity can be no farther increased, or m = w, that is ; and hence is the said uniform or greatest velocity to which the body may attain; which is evidently the greater in the subduplicate proportion of v the diameter of the ball. But v is always = √4gfs, the velocity generated by any accelerative force f in describing the space s; which being compared with the former, it gives s = (4/3)d, when f is = (N - n)/n; that is, the greatest velocity is that which is generated by the accelerating force (N - n)/n in passing over the space (4/3)d or 4/3 of the diameter of the ball, or it is equal to the velocity generated by gravity in describing the space . For ex. if the ball be of lead, which is about 11 1/4 times the density of water; then , and ; that is, the uniform or greatest velocity of a ball of lead, descending in water, is equal to that which a heavy body acquired by falling in vacuo through a space equal to 13 2/3 of the diameter of the ball, which velocity is nearly, or 8 times the root of the same space.
Hence it appears, how soon small bodies come to their greatest or uniform velocity in descending in a fluid, as water, and how very small that velocity is: which explains the reason of the slow precipitation of mud, and small particles, in water, as also why, in precipitations, the larger and gross particles descend soonest, and the lowest.
Farther, where N = n, or the density of the fluid is equal to that of the body, then N - n = 0, consequently the velocity and distance descended are each nothing, and the body will just float in any part of the fluid.
Moreover, when the body is lighter than the fluid, then N is less than n, and N - n becomes a negative quantity, or the force and motion tend the contrary way, that is, the ball will ascend up towards the top of the fluid by a motive force which is as n - N. In this case then, the body ascending by the action of the fluid, is moved exactly by the same laws as a heavier body falling in the fluid. Wherever the body is placed, it is sustained by the fluid, and carried up with a force equal to the difference of the weight of a quantity of the fluid of the same bulla as the body, from the weight of the body; there is therefore a force which continually acts equably upon the body; by which not only the action of gravity of the body is counteracted, so as that it is not to be considered in this case; but the body is also carried upwards by a motion equably accelerated, in the same manner as a body heavier than a fluid descends by its relative gravity: but the equability of acceleration is destroyed in the same manner by the Resistance, in the ascent of a body lighter than the fluid, as it is destroyed in the descent of a body that is heavier.
For the circumstances of the correspondent velocity, space, and time, &c, of a body moving in a fluid in which it is projected with a given velocity, or descending by its own weight, &c, see my Select Exercises, prop. 29, 30, 31, and 32, pag. 221 &c.
Resistance of the Air, in Pneumatics, is the force with which the motion of bodies, particularly of projectiles, is retarded by the opposition of the air or atmosphere. See Gunnery, Projectiles, &c.
The air being a fluid, the general laws of the Resistance of fluids obtain in it; subject only to some variations and irregularities from the different degrees of density in the different stations or regions of the atmosphere.
The Resistance of the air is chiefly of use in military projectiles, in order to allow for the differences caused in their flight and range by it. Before the time of Mr. Robins, it was thought that this Resistance to the motion of such heavy bodies as iron balls and shells, was too inconsiderable to be regarded, and that the rules and conclusions derived from the common parabolic theory, were sufficiently exact for the common practice of gunnery. But that gentleman shewed, in his New Principles of Gunnery, that, so far from being inconsiderable, it is in reality enormously great, and by no means to be rejected without incurring the grossest errors; so much so, that balls or shells which range, at the most, in the air, to the distance of two or three miles, would in a vacuum range to 20 or 30 miles, or more. To determine the quantity of this Resistance, in the case of different velocities, Mr. Robins discharged musket balls, with various degrees of known velocity, against his ballistic pendulums, placed at several different distances, and so discovered by experiment the quantity of velocity lost, when passing through those distances | or spaces of air, with the several known degrees of celerity. For having thus known, the velocity lost or destroyed, in passing over a certain space, in a certain time, (which time is very nearly equal to the quotient of the space divided by the medium velocity between the greatest and least, or between the velocity at the mouth of the gun and that at the pendulum); that is, knowing the velocity v, the space s, and time t, the resisting force is thence easily known, being equal to (vb)/(2gt) or (vVb)/(2gs), where b denotes the weight of the ball, and V the medium velocity above mentioned. The balls employed upon this occasion by Mr. Robins, were leaden ones, of 1/12 of a pound weight, and 3/4 of an inch diameter; and to the medium velocity of
| 1600 feet the Resistance was | 11 lb, |
| 1065 feet " it was | 2 4/5; |
Mr. Robins also invented another machine, having a whirling or circular motion, by which he measured the Resistances to larger bodies, though with much smaller velocities: it is described, and a figure of it given, near the end of the 1st vol. of his works.
That this resisting power of the air to swift motions is very sensibly increased beyond what Newton's theory for slow motions makes it, seems hence to be evident. By other experiments it appears that the Resistance is very sensibly increased, even in the velocity of 400 feet. However, this increased power of Resistance diminishes as the velocity of the resisted body diminishes, till at length, when the motion is sufficiently abated, the actual Resistance coincides with that supposed in the theory nearly. For these varying Resistances Mr. Robins has given a rule, extending to 1670 feet velocity.
Mr. Euler has shewn, that the common doctrine of Resistance answers pretty well when the motion is not very swift, but in swift motions it gives the Resistance less than it ought to be, on two accounts. 1. Because in quick motions, the air does not fill up the space behind the body fast enough to press on the hinder parts, to counterbalance the weight of the atmosphere on the fore part. 2. The density of the air before the ball being increased by the quick motion, will press more strongly on the fore part, and so will resist more than lighter air in its natural state. He has shewn that Mr. Robins has restrained his rule to velocities not exceeding 1670 feet per second; whereas had he extended it to greater velocities, the result must have been erroneous; and he gives another formula himself, and deduces conclusions differing from those of Mr. Robins. See his Principles of Gunnery investigated, translated by Brown in 1777, pa. 224 &c.
Mr. Robins having proved that, in very great changes of velocity, the Resistance does not accurately follow the duplicate ratio of the velocity, lays down two positions, which he thought might be of some service in the practice of artillery, till a more complete and accurate theory of Resistance, and the changes of its augmentation, may be obtained. The first of these is, that till the velocity of the projectile surpass 1100 or 1200 feet in a second, the Resistance may be esteemed to be in the duplicate ratio of the velocity; and the second is, that when the velocity exceeds 1100 or 1200 feet, then the absolute quantity of the Resistance will be near 3 times as great as it should be by a comparison with the smaller velocities. Upon these principles he proceeds in approximating to the actual ranges of pieces with small angles of elevation, viz, such as do not exceed 8° or 10°, which he sets down in a table, compared with their corresponding potential ranges. See his Mathematical Tracts, vol. 1 pa. 179 &c. But we shall see presently that these positions are both without foundation; that there is no such thing as a sudden or abrupt change in the law of Resistance, from the square of the velocity to one that gives a quantity three times as much; but that the change is slow and gradual, continually from the smallest to the highest velocities; and that the increased real Resistance no where rises higher than to about double of that which Newton's theory gives it.
Mr. Glenie, in his History of Gunnery, 1776, pa. 49, observes, in consequence of some experiments with a rifled piece, properly fitted for experimental purposes, that the Resistance of the air to a velocity somewhat less than that mentioned in the first of the above propositions, is considerably greater than in the duplicate ratio of the velocity; and that, to a celerity somewhat greater than that stated in the second, the Resistance is considerably less than that which is treble the Resistance in the said ratio. Some of Robins's own experiments seem necessarily to make it so; since, to a velocity no quicker than 400 feet in a second, he found the Resistance to be somewhat greater than in that ratio. But the true value of the ratio, and other circumstances of this Resistance, will more fully appear from what follows.
The subject of the Resistance of the air, as begun by Robins, has been prosecuted by myself, to a very great extent and variety, both with the whirling machine, and with cannon balls of all sizes, from 1 lb to 6 lb weight, as well as with figures of many other different shapes, both on the fore part and hind part of them, and with planes set at all varieties of angles of inclination to the path or motion of the same; from all which I have obtained the real Resistance to bodies for all velocities, from 1 up to 2000 feet per second; together with the law of the Resistance to the same body for all different velocities, and for different sizes with the same velocity, and also for all angles of inclination; a full account of which would make a book of itself, and must be reserved for some other occasion. In the mean time, some general tables of conclusions may be taken as below. |
| Smad | Large Hemis. | Cone | Resif. | |||||
| Veloc. | Hemis. | Cylin- | Whole | as the | ||||
| per | der | globe | power | |||||
| See | <*>at | flat | round | vertex | base | of the | ||
| side | side | side | vrloe. | |||||
| feet | oz | oz | oz | oz | oz | oz | oz | |
| 3 | .028 | .051 | .020 | .028 | .064 | .050 | .027 | |
| 4 | .048 | .096 | .039 | .048 | .109 | .090 | .047 | |
| 5 | .072 | .148 | .063 | .071 | .162 | .143 | .068 | |
| 6 | .103 | .211 | .092 | .098 | .225 | .205 | .094 | |
| 7 | .141 | .284 | .123 | .129 | .298 | .278 | .125 | |
| 8 | .184 | .368 | .160 | .168 | .382 | .360 | .162 | |
| 9 | .233 | .464 | .199 | .211 | .478 | .456 | .205 | |
| 10 | .287 | .573 | .242 | .260 | .587 | .565 | .255 | |
| 11 | .349 | .698 | .292 | .315 | .712 | .688 | .310 | 2.052 |
| 12 | .418 | .836 | .347 | .376 | .850 | .826 | .370 | 2.042 |
| 13 | .492 | .988 | .409 | .440 | 1.000 | .979 | .435 | 2.036 |
| 14 | .573 | 1.154 | .478 | .512 | 1.166 | 1.145 | .595 | 2.031 |
| 15 | .661 | 1.336 | .552 | .589 | 1.346 | 1.327 | .581 | 2.031 |
| 16 | .754 | 1.538 | .634 | .673 | 1.546 | 1.526 | .663 | 2.033 |
| 17 | .853 | 1.757 | .722 | .762 | 1.763 | 1.745 | .752 | 2.038 |
| 18 | .959 | 1.998 | .818 | .858 | 2.002 | 1.986 | .848 | 2.044 |
| 19 | 1.073 | 2.258 | .922 | .959 | 2.260 | 2.246 | .949 | 2.047 |
| 20 | 1.196 | 2.542 | 1.033 | 1.060 | 2.540 | 2.528 | 1.057 | 2.051 |
| Mean | ||||||||
| pro<*>or. | 140 | 288 | 119 | 126 | 291 | 285 | 124 | 2.040 |
| No<*>. | ||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
In this Table are contained the Resistances to several forms of bodies, when moved with several degrees of velocity, from 3 feet per second to 20. The names of the bodies are at the tops of the columns, as also which end went foremost through the air; the different velocities are in the first column, and the Resistances on the same line, in their several columns, in avoirdupois ounces and decimal parts. So on the first line are contained the Resistances when the bodies move with a velocity of 3 feet in a second, viz, in the 2d column for the small hemisphere, of 4 3/4 inches diameter, its Resistance .028 oz when the flat side went foremost; in the 3d and 4th columns the Resistances to a larger hemisphere, first with the flat side, and next the round side foremost, the diameter of this, as well as all the following figures being 6 5/8 inches, and therefore the area of the great circle = 32 sq. inches, or 2/9 of a sq. foot; then in the 5th and 6th columns are the Resistances to a cone, first its vertex and then its base foremost, the altitude of the cone being 6 5/8 inches, the same as the diameter of its base; in the 7th column the Resistance to the end of the cylinder, and in the 8th that against the whole globe or sphere. All the numbers show the real weights which are equal to the Resistances; and at the bottoms of the columns are placed proportional numbers, which shew the mean proportions of the Resistances of all the figures to one another, with any velocity. Lastly, in the 9th column are placed the exponents of the power of the velocity which the Resistances in the 8th column bear to each other, viz, which that of the 10 feet velocity bears to each of the following ones, the medium of all of them being as the 2.04 power of the velocity, that is, very little above the square or second power of the velocity, so far as the velocities in this Table extend.
From this Table the following inferences are easily deduced.
1. That the Resistance is nearly in the same proportion as the surfaces; a small increase only taking place in the greater surfaces, and for the greater velocities. Thus, by comparing together the numbers in the 2d and 3d columns, for the bases of the two hemispheres, the areas of which bases are in the proportion of 17 3/4 to 32, or 5 to 9 very nearly, it appears that the numbers in those two columns, expressing the Resistances, are nearly as 1 to 2 or 5 to 10, as far as the velocity of 12 feet; but after that, the Resistances on the greater surface increase gradually more and more above that proportion.
2. The Resistance to the same surface, with different velocities, is, in these slow motions, nearly as the square of the velocity; but gradually increases more and more above that proportion as the velocity increases. This is manifest from all the columns; and the index of the power of the velocity is set down in the 9th column, for the Resistances in the 8th, the medium being 2.04; by which it appears that the Resistance to the same body is, in these slow motions, as the 2.04 power of the velocity, or nearly as the square of it.
3. The round ends, and sharp ends, of solids, suffer less Resistance than the flat or plane ends, of the same diameter; but the sharper end has not always the less Resistance. Thus, the cylinder, and the flat ends of the hemisphere and cone, have more Resistance, than the round or sharp ends of the same; but the round side of the hemisphere has less Resistance than the sharper end of the cone.
4. The Resistance on the base of the hemisphere, is to that on the round, or whole sphere, as 2 1/3 to 1, instead of 2 to 1, as the theory gives that relation. Also the experimented Resistance, on each of these, is nearly 1/4 more than the quantity assigned by the theory.
5. The Resistance on the base of the cone, is to that on the vertex, nearly as 2 3/10 to 1; and in the same ratio is radius to the sine of the angle of inclination of the side of the cone to its path or axis. So that, in this instance, the Resistance is directly as the sine of the angle of incidence, the transverse section being the same.
6. When the hinder parts of bodies are of different forms, the Resistances are different, though the foreparts be exactly alike and equal; owing probably to the different pressures of the air on the hinder parts. Thus, the Resistance to the fore part of the cylinder, is less than on the equal flat surface of the cone, or of the hemisphere; because the hinder part of the cylinder is more pressed or pushed, by the following air than those of the other two figures; also, for the same reason, the base of the hemisphere suffers a less Resistance than that of the cone, and the round side of the hemisphere less than the whole sphere. |
| Veloc. per | Resist. by | Resist. by | Ratio of | Resist. as |
| Exper. | Theory. | Exper. to | the power | |
| sec. in feet. | oz. | oz. | Theory. | of the veloc. |
| 5 | 0.006 | 0.005 | 1.20 | |
| 10 | 0.024 1/2 | 0.020 | 1.23 | |
| 15 | 0.055 | 0.044 | 1.25 | |
| 20 | 0.100 | 0.079 | 1.27 | |
| 25 | 0.157 | 0.123 | 1.28 | 2.022 |
| 30 | 0.23 | 0.177 | 1.30 | 2.055 |
| 40 | 0.42 | 0.314 | 1.33 | 2.068 |
| 50 | 0.67 | 0.491 | 1.36 | 2.075 |
| 100 | 2.72 | 1.964 | 1.38 | 2.059 |
| 200 | 11 | 7.9 | 1.40 | 2.041 |
| 300 | 25 | 18.7 | 1.41 | 2.039 |
| 400 | 45 | 31.4 | 1.43 | 2.039 |
| 500 | 72 | 49 | 1.47 | 2.044 |
| 600 | 107 | 71 | 1.51 | 2.051 |
| 700 | 151 | 96 | 1.57 | 2.059 |
| 800 | 205 | 126 | 1.63 | 2.067 |
| 900 | 271 | 159 | 1.70 | 2.077 |
| 1000 | 350 | 196 | 1.78 | 2.086 |
| 1100 | 442 | 238 | 1.86 | 2.095 |
| 1200 | 546 | 283 | 1.90 | 2.102 |
| 1300 | 661 | 332 | 1.99 | 2.107 |
| 1400 | 785 | 385 | 2.04 | 2.111 |
| 1500 | 916 | 442 | 2.07 | 2.113 |
| 1600 | 1051 | 503 | 2.09 | 2.113 |
| 1700 | 1186 | 568 | 2.08 | 2.111 |
| 1800 | 1319 | 636 | 2.07 | 2.108 |
| 1900 | 1447 | 709 | 2.04 | 2.104 |
| 2000 | 1569 | 786 | 2.00 | 2.098 |
In the first column of this Table are contained the several velocities, gradually from o up to the great velocity of 2000 feet per second, with which a ball or globe moved. In the 2d column are the experimented Resistances, in averdupois ounces. In the 3d column are the correspondent Resistances, as computed by the foregoing theory. In the 4th column are the ratios of these two Resistances, or the quotients of the former divided by the latter. And in the 5th or last, the indexes of the power of the velocity which is proportional to the experimented Resistance; which are found by comparing the Resistance of 20 feet velocity with each of the following ones.
From the 2d, 3d and 4th columns it appears, that at the beginning of the motion, the experimented Resistance is nearly equal to that computed by theory; but that, as the velocity increases, the experimented Resistance gradually exceeds the other more and more, till at the velocity of 1300 feet the former becomes just double the latter; after which the difference increases a little farther, till at the velocity of 1600 or 1700, where that excess is the greatest, and is rather less than 2 1/10; after this, the difference decreases gradually as the velocity increases, and at the velocity of 2000, the former Resistance again becomes just double the latter.
From the last column it appears that, near the begin- ning, or in slow motions, the Resistances are nearly as the square of the velocities; but that the ratio gradually increases, with some small variation, till at the velocity of 1500 or 1600 feet it becomes as the 2 1/9 power of the velocity nearly, which is its highest ascent; and after that it gradually decreases again, as the velocity goes higher. And similar conclusions have also been derived from experiments with larger balls or globes.
And hence we perceive that Mr. Robins's positions are erroneous on two accounts, viz, both in stating that the Resistance changes suddenly, or all at once, from being as the square of the velocity, so as then to become as some higher and constant power; and also when he states the Resistance as rising to the height of 3 times that which is given by the theory: since the ratio of the Resistance both increases gradually from the beginning, and yet never ascends higher than 2 9/100 of the theory.
| Table III. Resistance to a Plane, set at various An- | |||
| gles of Inclination to its Path. | |||
| Angle with the | Experim. Re- | Resist. by this | Sines of the An- |
| sistances. | Formula. | gles to Radius | |
| Path. | oz. | .84s1.342c | .840. |
| 0° | .000 | .000 | .000 |
| 5 | .015 | .009 | .073 |
| 10 | .044 | .035 | .146 |
| 15 | .082 | .076 | .217 |
| 20 | .133 | .131 | .287 |
| 25 | .200 | .199 | .355 |
| 30 | .278 | .278 | .420 |
| 35 | .362 | .363 | .482 |
| 40 | .448 | .450 | .540 |
| 45 | .534 | .535 | .594 |
| 50 | .619 | .613 | .643 |
| 55 | .684 | .680 | .688 |
| 60 | .729 | .736 | .727 |
| 65 | .770 | .778 | .761 |
| 70 | .803 | .808 | .789 |
| 75 | .823 | .826 | .811 |
| 80 | .835 | .836 | .827 |
| 85 | .839 | .839 | .838 |
| 90 | .840 | .840 | .840 |
In the 2d column of this Table are contained the actual experimented Resistances, in ounces, to a plane of 32 square inches, or 2/9 of a square foot, moved through the air with a velocity of exactly 12 feet per second, when the plane was set so as to make, with the direction of its path, the corresponding angles in the first column.
And from these I have deduced this formula, or theorem, viz, .84s1.842c, which brings out very nearly the same numbers, and is a general theorem for every angle, for the same plane of 2/9 of a foot, and moved with the same velocity of 12 feet in a second of time; where s is the sine, and c the cosine of the angles of inclination in the first column. |
If a theorem be desired for any other velocity v, and any other plane whose area is a, it will be this: (1/38)av2s1.842c, or more nearly (1/42)av2.04s1.842c; which denotes the Resistance nearly to any plane surface whose area is a, moved through the air with the velocity v, in a direction making with that plane an angle, whose sine is s, and cosine c.
If it be water or any other fluid, different from air, this formula will be varied in proportion to the density of it.
By this theorem were computed the numbers in the 3d column; which it is evident agree very nearly with the experiment Resistances in the 2d column, excepting in two or three of the small numbers near the beginning, which are of the least consequence. In all other cases, the theorem gives the true Resistance very nearly. In the 4th or last column are entered the sines of the angles of the first column, to the radius .84, in order to compare them with the Resistances in the other columns. From whence it appears, that those Resistances bear no sort of analogy to the sines of the angles, nor yet to the squares of the sines, nor to any other power of them whatever. In the beginning of the columns, the sines much exceed the Resistances all the way till the angle be between 55 and 60 degrees; after which the sines are less than the Resistances all the way to the end, or till the angle become of 90 degrees.
Mr. James Bernoulli gave some theorems for the Resistances of different figures, in the Acta Erud. Lips. for June 1693, pa. 252 &c. But as these are deduced from theory only, which we find to be so different from experiment, they cannot be of much use. Messieurs Euler, D'Alembert, Gravesande, and Simpson, have also written pretty largely on the theory of Resistances, besides what had been done by Newton.
Solid of Least Resistance. Sir Isaac Newton, from his general theory of Resistance, deduces the figure of a solid which shall have the least Resistance of the same base, height and content.
The figure is this. Suppose DNG to be a curve of such a nature, that if from any point N the ordinate NM be drawn perpendicular to the axis AB; and from a given point G there be drawn GR parallel to a tangent at N, and meeting the axis produced in R; then if MN be to GR, as GR3 to 4BR X BG2, a solid described by the revolution of this figure about its axis AB, moving in a medium from A towards B, is less resisted than any other circular solid of the same base, &c.
This theorem, which Newton gave without a demonstration, has been demonstrated by several mathe- maticians, as Facio, Bernoulli, Hospital, &c. See Maclaurin's Flux. sect. 606 and 607; also Horslev's edit. of Newton, vol. 2, pag. 390. See also Act. Erud. 1699, pa. 514; and Mem. de l'Acad. &c; also Robins's View of Newton's method for comparing the Resistance of Solids, 8vo, 1734; and Simpson's Fluxions, art. 413; or my Principles of Bridges, prop. 11 and 12.
M. Bouguer has resolved this problem in a very general manner; not in supposing the solid to be formed by a revolution, of any figure whatever. The problem, as enunciated and resolved by M. Bouguer, is this: Any base being given, to find what kind of solid must be formed upon it, so that the impulse upon it may be the least possible. Properly however it ought to be the retardive force, or the impulse divided by the weight or mass of matter in the body, that ought to be the minimum.
